The long-term goal, of this proposed exploratory research program, is to develop imaging technologies for diagnostic purposes that can identify, with a high resolution, small abnormalities in tissue. The gain will be diagnoses that can result in more successful treatments. Our data will be the mechanical transient response of tissue, measured by ultrasonic devices, and subject to loading on the boundary of the tissue. Instead of using the tissue displacement and its derivatives to create images, our aim is to reconstruct the elastic shear stiffness, the quantity that characterizes the body's resistance to shape change. To do this we solve an illposed inverse problem Our Hypothesis is that: Novel methods and algorithms using inverse problem solving techniques, can be used to reconstruct the shear stiffness, with a higher degree of resolution that currently available methods, in 3-D from ultrasonic measurements of the displacement field. Our Research Objectives are: (1) Year 1 Objectives: analyze the arrival time algorithm to determine it's strengths and overcome weaknesses; test level set methods and associated total variation techniques; implement detailed forward model for sensitivity analysis and simulated data generation; derive finite element based formulations for handling inverse problem; (2) Year 2 Objectives: finish implementation of algorithms; test and evaluate algorithms on simulated and real data; explore and utilize fast algorithms and investigate parallel implementation; draw conclusions regarding quality of results and strengths of the methods.